"When you have a squad as small as ours it's vital to fill the gaps when players leave. We have an experienced midfielder in Lee Carsley but with Fab gone we need to further strengthen in that department."
Birmingham's managing director Karren Brady demanded a deal which could eventually be worth £5.75 million.
Arsenal have also received money from the deal.
With add-ons, Muamba's move to Birmingham earned them £3.15 million and they will also receive around £600,000 which is 30 per cent of the profit made on the Bolton transfer.